# Explicit and implicit methods

## Approaches for approximating solutions to differential equations / From Wikipedia, the free encyclopedia

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**Explicit and implicit methods** are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. *Explicit methods* calculate the state of a system at a later time from the state of the system at the current time, while *implicit methods* find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if $Y(t)$ is the current system state and $Y(t+\Delta t)$ is the state at the later time ($\Delta t$ is a small time step), then, for an explicit method

- $Y(t+\Delta t)=F(Y(t))\,$

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while for an implicit method one solves an equation

- $G{\Big (}Y(t),Y(t+\Delta t){\Big )}=0\qquad (1)\,$

to find $Y(t+\Delta t).$